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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlimddvcbv | Structured version Visualization version GIF version |
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40911. The equivalent of this theorem without the bound variable change is rexlimddv 3250. Usage of this theorem is discouraged because it depends on ax-13 2379, see rexlimddvcbvw 40912 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rexlimddvcbv.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) |
rexlimddvcbv.2 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) |
rexlimddvcbv.3 | ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexlimddvcbv | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimddvcbv.1 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) | |
2 | rexlimddvcbv.3 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) | |
3 | rexlimddvcbv.2 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) | |
4 | 2, 3 | rexlimdvaacbv 40911 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓)) |
5 | 1, 4 | mpd 15 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 |
This theorem is referenced by: (None) |
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